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Introduction Survival times are data that measure follow-up time from a defined starting point to the occurrence of a given event, for example the time from the beginning to the end of a

Introduction

Survival times are data that measure follow-up time from a

defined starting point to the occurrence of a given event, for

example the time from the beginning to the end of a remission

period or the time from the diagnosis of a disease to death

Standard statistical techniques cannot usually be applied

because the underlying distribution is rarely Normal and the

data are often ‘censored’ A survival time is described as

censored when there is a follow-up time but the event has not

yet occurred or is not known to have occurred For example, if

remission time is being studied and the patient is still in

remission at the end of the study, then that patient’s

remission time would be censored If a patient for some

reason drops out of a study before the end of the study

period, then that patient’s follow-up time would also be

considered to be censored

The hypothetical data set given in Table 1 will be used for

illustrative purposes in this review For this data set the event

is the death of the patient, and so the censored data are

those where the outcome is survived or unknown

Estimating the survival curve using the

Kaplan–Meier method

In analyzing survival data, two functions that are dependent

on time are of particular interest: the survival function and the

hazard function The survival function S(t) is defined as the

probability of surviving at least to time t The hazard function

h(t) is the conditional probability of dying at time t having

survived to that time

The graph of S(t) against t is called the survival curve The Kaplan–Meier method can be used to estimate this curve from the observed survival times without the assumption of an underlying probability distribution The method is based on the basic idea that the probability of surviving k or more periods from entering the study is a product of the k observed survival rates for each period (i.e the cumulative proportion surviving), given by the following:

S(k) = p1× p2× p3× … × pk Here, p1 is the proportion surviving the first period, p2is the proportion surviving beyond the second period conditional on having survived up to the second period, and so on The proportion surviving period i having survived up to period i is given by:

Where riis the number alive at the beginning of the period and dithe number of deaths within the period

To illustrate the method the data for the patients receiving treatment 2 from Table 1 will be used The survival times, including the censored values (indicated by + in Table 2), must be ordered in increasing duration If a censored time has the same value as an uncensored time, then the uncensored should precede the censored The calculations are shown in Table 2 Where there is a censored time the proportion surviving will be 1 This does not alter the

Review

Statistics review 12: Survival analysis

1Senior Lecturer, School of Computing, Mathematical and Information Sciences, University of Brighton, Brighton, UK

2Senior Registrar in ICU, Liverpool Hospital, Sydney, Australia

Correspondence: Viv Bewick, v.bewick@brighton.ac.uk

Published online: 6 September 2004 Critical Care 2004, 8:389-394 (DOI 10.1186/cc2955)

This article is online at http://ccforum.com/content/8/5/389

© 2004 BioMed Central Ltd

Abstract

This review introduces methods of analyzing data arising from studies where the response variable is

the length of time taken to reach a certain end-point, often death The Kaplan–Meier methods, log

rank test and Cox’s proportional hazards model are described

Keywords Cox’s proportional-hazards model, cumulative hazard function H(t), hazard ratio, Kaplan–Meier method,

log rank test, survival function S(t)

i i i i

r d r

cumulative proportion surviving, and so these calculations

can be omitted from the table For more detailed explanation,

see Swinscow and Campbell [1]

Plotting the cumulative proportion surviving against the

survival times gives the stepped survival curve shown in Fig 1

This method is found in most statistical packages Figure 2 is

the output from a statistical package used to compare the

survival curves for the two treatment groups for the data given in Table 1

It can be seen that patients on treatment 1 appear to have a higher survival rate than those on treatment 2 The graph can

be used to estimate the median survival time because this is the time with probability of survival of 0.5 The median survival time for those on treatment 2 appears to be 5 days versus about 37 days on treatment 1

Comparing survival curves of two groups using the log rank test

Comparison of two survival curves can be done using a statistical hypothesis test called the log rank test It is used to

Table 1

Survival time, age and outcome for a group of patients

diagnosed with a disease and receiving one of two treatments

number (days) Outcome Treatment (years)

Table 2

Calculations for the Kaplan–Meier estimate of the survival function for the treatment 2 data from Table 1

Patient number (days) to be alive (ri) Deaths (di) surviving (pi) surviving (S[t])

Figure 1

Plot of the survival curve for treatment 2

Survival time (days)

30 20

10 0

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Censored +

test the null hypothesis that there is no difference between

the population survival curves (i.e the probability of an event

occurring at any time point is the same for each population)

The test statistic is calculated as follows:

Where the O1 and O2 are the total numbers of observed events in groups 1 and 2, respectively, and E1 and E2the total numbers of expected events

The total expected number of events for a group is the sum of the expected number of events at the time of each event The expected number of events at the time of an event can be calculated as the risk for death at that time multiplied by the number alive in the group Under the null hypothesis, the risk

of death (number of deaths/number alive) can be calculated from the combined data for both groups Table 3 shows the calculation of the expected number of deaths for treatment group 2 for the example data For example, at the beginning

of day 4 when the third death (event 3) takes place, there are

13 patients still alive One dies, giving a risk for death of 1/13 = 0.077 Six of the 13 patients are from treatment group 2, and therefore the expected number of deaths is given by 6 × 0.077 = 0.46 at event 3 The total expected number of events for group 2 is calculated as:

Where r2i is the number alive from group 2 at the time of event i E1can be calculated as n – E2, where n is the total number of events

The test statistic is compared with a χ2 distribution with 1 degree of freedom It is a simplified version of a statistic that

is often calculated in statistical packages [2]

Figure 2

Survival curves for the two treatment groups for the data in Table 1

Survival time (days)

80 70 60 50 40 30 20 10

0

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Treatment 2 Treatment 1 Censored +

2

2 2 2 1

2 1 1 2

E ) E (O E

) E (O rank) (log

Table 3

Calculations for the log-rank test to compare treatments for the data in Table 1

time Treatment known to Deaths Risk for death to be alive from of events in treatment

(days) group be alive (ri) (di) (di/ri) treatment group 2 (r2i) group 2 (E2i)

0

E2= 2.92

∑=

= k

1 i i 2 i

i

r

d E

χ

For the data in Table 1, the total number of expected deaths

for treatment group 2 is calculated as 2.92 and the total

number of observed deaths is 10, giving a total number of

expected deaths for treatment group 1 of 10 – 2.92 = 7.08

The value of the test statistic is therefore calculated as

follows:

This gives a P value of 0.032, which indicates a significant

difference between the population survival curves

An assumption for the log rank test is that of proportional

hazards This is discussed below Small departures from this

assumption, however, do not invalidate the test

Cox’s proportional hazards model (Cox

regression)

The log rank test is used to test whether there is a difference

between the survival times of different groups but it does not

allow other explanatory variables to be taken into account

Cox’s proportional hazards model is analogous to a multiple

regression model and enables the difference between

survival times of particular groups of patients to be tested

while allowing for other factors In this model, the response

(dependent) variable is the ‘hazard’ The hazard is the

probability of dying (or experiencing the event in question)

given that patients have survived up to a given point in time,

or the risk for death at that moment

In Cox’s model no assumption is made about the probability

distribution of the hazard However, it is assumed that if the

risk for dying at a particular point in time in one group is, say,

twice that in the other group, then at any other time it will still

be twice that in the other group In other words, the hazard

ratio does not depend on time

The model can be written as:

ln h(t) = ln h0(t) + b1x1+ … + bpxp

or ln = b1x1+ … + bpxp

Where h(t) is the hazard at time t; x1, x2 … xp are the

explanatory variables; and h0(t) is the baseline hazard when all

the explanatory variables are zero The coefficients b1, b2… bp

are estimated from the data using a statistical package

Because hazard measures the instantaneous risk for death, it

is difficult to illustrate it from sample data Instead, the

cumulative hazard function H(t) can be examined This can be

obtained from the cumulative survival function S(t) as follows:

H(t) = –ln S(t)

The estimated cumulative hazard function for the example data given in Table 1 is shown in Table 4

The assumption that the proportional hazards stay constant over time can be inspected by looking at a graph showing the logarithm of the estimated cumulative hazard function The assumption is equivalent to assuming that the difference between the logarithms of the hazards for the two treatments does not change with time, or equally that the difference between the logarithms of the cumulative hazard functions is constant Figure 3 is the graph for the example data The lines for the two treatments are roughly parallel, suggesting that the proportional hazards assumption is reasonable in this case A more formal test of the assumption is possible (see Armitage and coworkers [2]) Note that, in this graph, the time scale was also logarithmically transformed This was to make the comparison clearer between the two treatments, but it does not affect the vertical positioning of the lines Cox’s regression was applied to the example data using treatment and age as explanatory variables The output is shown in Table 5

The P values indicate that the difference between treatments

was bordering on statistical significance, whereas there was

59 4 92 2

) 92 2 6 ( 08 7

) 08 7 4

=

− +

−

(t) h h(t) 0

Table 4 Cumulative hazard functions (logarithmic scale) for the example data

Survival time Cumulative survival: Cumulative hazard:

Treatment 1

Treatment 2 1

strong evidence that age was associated with length of

survival The coefficient for treatment, –1.887, is the

logarithm of the hazard ratio for a patient given treatment 1

compared with a patient given treatment 2 of the same age

The exponential (antilog) of this value is 0.152, indicating that

a person receiving treatment 1 is 0.152 times as likely to die

at any time as a patient receiving treatment 2; that is, the risk

associated with treatment 1 appears to be much lower

However, the confidence interval contains 1, indicating that

there may be no difference in risk associated with the two

treatments

Using the Kaplan–Meier (log rank) test, the P value for the

difference between treatments was 0.032, whereas using

Cox’s regression, and including age as an explanatory

variable, the corresponding P value was 0.052 This is not a

substantial change and still suggests that a difference

between treatments is likely In this case age is clearly an

important explanatory variable and should be included in the

analysis

The exponential of the coefficient for age, 1.247, indicates

that a patient 1 year older than another patient, both being

given the same treatment, has an increased risk for dying, by

a factor of 1.247 Note that, in this case, the confidence interval does not contain 1, indicating the statistical significance of age

Further models for survival data, allowing for different assumptions, are discussed by Kirkwood and Sterne [3]

An example from the literature

Dupont and coworkers [4] investigated the survival of patients with bronchiectasis according to age and use of long-term oxygen therapy The Kaplan–Meier curves and results of the log rank tests shown in Fig 4 indicate that there

is a significant difference between the survival curves in each case

The authors also applied Cox’s proportional hazards analysis and obtained the results given in Table 6 These results indicate that both age and long-term oxygen therapy have a significant effect on survival The estimated risk ratio for age, for example, suggests that the risk for death for patients over the age of 65 years is 2.7 times greater than that for those below 65 years

Figure 3

Cumulative hazard functions for the example data

Survival time (days) (log scale)

100 10

1

1.50

1.00

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.15

Treatment 1 2

Table 5

Application of Cox’s regression to the example data, using

treatment and age as explanatory variables

95.0%

Coefficient Standard confidence

(b) error P eb interval for eb

Treatment –1.887 0.973 0.052 0.152 0.022–1.020

Age 0.220 0.085 0.010 1.247 1.054–1.474

Figure 4

The Kaplan–Meier estimates of survival for (a) age > 65 years or

≤65 years, and (b) long-term oxygen therapy (LTOT) before intensive

care unit admission (yes/no) The P values are for the log rank test.

Assumptions and limitations

The log rank test and Cox’s proportional hazards model assume that the hazard ratio is constant over time Care must

be taken to check this assumption

Conclusion

Survival analysis provides special techniques that are required to compare the risks for death (or of some other event) associated with different treatments or groups, where the risk changes over time In measuring survival time, the start and end-points must be clearly defined and the censored observations noted Only the most commonly used techniques are introduced in this review Kaplan–Meier provides a method for estimating the survival curve, the log rank test provides a statistical comparison of two groups, and Cox’s proportional hazards model allows additional covariates

to be included Both of the latter two methods assume that the hazard ratio comparing two groups is constant over time

Competing interests

The authors declare that they have no competing interests

References

1 Swinscow TDV, Campbell MJ: Statistics at Square One London:

BMJ Books; 2002

2 Armitage P, Berry G, Matthews JNS: Statistical Methods in

Medical Research, 4th edn Oxford, UK: Blackwell Science;

2002

3 Kirkwood BR, Sterne JAC: Essential Medical Statistics, 2nd edn.

Oxford, UK: Blackwell Science Ltd; 2003

4 Dupont M, Gacouin A, Lena H, Lavoue S, Brinchault G, Delaval P,

Thomas R: Survival of patients with bronchiectasis after the

first ICU stay for respiratory failure Chest 2004,

125:1815-1820

Table 6

Results of Cox’s proportional hazards analysis for the patients with bronchiectasis

Age (>65 years) 2.7 1.15–6.29 0.022

LTOT, long-term oxygen therapy